- Email: [email protected]
On the rms-radius of the deuteron
.__ __ lli!B
cm
16 May 1996
PHYSICS
ELSEVIER
LETTERS B
Physics Letters B 375 ( 1996) 16-20
On the rms-radius of the deuteron I. Sick ‘, D. Trautmann Dept. fiir Physik und Astronomie, Universitiit Basel. CH-4056 Basel, Switzerland Received
16 January
1996; revised manuscript received 16 February Editor: R.H. Siemssen
1996
Abstract We study the world data on elastic electron-deuteron scattering in order to determine the deuteron charge rms-radius. After accounting for the Coulomb distortion we find a radius of 2.128f 0.011 fm, which is in perfect agreement with the
radii deduced from other sources. This removes a disturbing discrepancy first pointed out by Klarsfeld et al. Keywords: Deuteron
radius; Electron scattering;
Coulomb
distortion
1. Introduction The deuteron is a fundamental system that has received extensive attention in the past, by both theory and experiment. The wave function of the deuteron can be calculated accurately for any given nucleonnucleon (N-N) potential. Many observables, and the electromagnetic form factors in particular, can be predicted with high precision. Overall, the studies of the past have shown that the information on the deuteron emerging from NN scattering and the one from electron scattering are compatible. The two types of observables then can be used as complementary sources of knowledge on the properties of the 2N-system. The work of Klarsfeld et al. [ 11, however, has revealed a disturbing discrepancy: The deuteron root-mean-square (rms) radius calculated using many N-N potentials shows a tight linear correlation with the N-N triplett scattering length predicted by the potential employed. For the well known experimental scattering length this linear ’ E-mail: [email protected]. 0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO370-2693(96)00214-6
relation misses the experimental deuteron rms-radius derived from electron scattering by 0.019 ho.003 fm. The very detailed treatise of Wong [2], which also gives reference to much of the previous work on the subject, has confirmed this disagreement For a system as well understood as the deuteron such a discrepancy is most disturbing. As a consequence, various authors have studied corrections that could remove this problem. In particular they have looked at the potential role played by an energydependence of the N-N interaction off-shell, and the role of non-nucleonic degrees of freedom. Mesonic [ 3,1,4J and quark [2,4] degrees have been investigated, but no explanation of the disagreement has been found; their effect is an order of magnitude too small. Recent very accurate data on the 2p-2s-transition energy in atomic deuterium [ 5,6] have drawn further attention to this question. Therms-radius derived from the atomic data is compatible with the one obtained from N-N scattering, but disagrees with the radius derived from electron scattering. In this Letter, we describe an alternative analysis of the electron scattering data on the deuteron which has
I. Sick, D. Trautmann/Physics
been performed in the hope to elucidate the origin of this puzzle.
2. Analysis In the present “model independent” analysis we write the elastic cross section in terms of the usual form factors F or structure functions A, B da(& 0) = ~Mott(Ev@)[A(q) + dQ
B(d$W/2>1
with the definition
+
Md ( xE.Ld)’ I’
B(q)
;+r&
= ( $%dj2
$T(
(9)
1+
rl)F;,(q)
with
I’ rl =
q2/(4MS)
and the corresponding equations for the tensor polarization observables [7]. In the above equations, M are the masses, ,u(Q) the deuteron magnetic (quadrupole) moment, and q is the four-momentum transfer. Of prime interest for this Letter is the low-q behaviour of FCQ N 1 - q2(r2)/6 + . . . . or the corresponding one for A(q) (corrected for the small FM] contribution using A (r*) = -&j/M;). We parameterize the deuteron form factors Fa (q) ,FM, (q) ,Fa (q) [or, alternatively, the structure functions A(q) and B(q)] using the sum-ofgaussians form factors SOG [ 81. This parameterization depends on one parameter, the width y of the gaussians which governs the large-q behaviour. This width is set to a value smaller than the size of the proton (we use 0.6 fm), as with increasing momentum transfer q the deuteron form factors are not expected to fall more slowly than the proton form factors, For the present study, only the behaviour of Fa (q) [or A(q) ] at small momentum transfer q is needed; the behaviour of the corresponding r-space densities a priori is of little interest. For a better understanding of a constraint we are going to use, it is useful, however, to also consider the SOG densities in r-space. (At low q the correspondence of density and form factor is well defined as the contribution of MEC is small).
Letters B 375 (1996116-20
17
The individual terms of the SOG basis correspond to gaussians of narrow width y placed at many different radii Ri. The deuteron wave function exhibits a long tail towards large I due to the weak binding of the deuteron, a tail that affects the shape of the form factors at very low q. As the shape of the tail at large r is well known - it depends only on the binding energy of the deuteron - this shape can be used as a constraint during the analysis of the data. This constraint amounts to fixing the ratio of the amplitudes of the gaussians for radii Ri 2 4 fm to the ratios given by a fit to the form factors or structure functions calculated from wave functions that have the correct binding energy. For the region r < 4 fm (where r refers to the radius relative to the center of mass of the deuteron) we use 10 parameters (gaussians) to describe each of the 3 (2) form factors. Various tests have been performed by fitting the form factors calculated by Henning et al. [9] for different N-N potentials. These tests show that the parameterization allows to fit the form factors up to the largest momentum transfers of the data, and they show that the fits accurately reproduce the rms-radii of the corresponding densities.
3. Data In the present work, we employ the world data on e-d scattering [ lo-341 ; relevant for a determination of the rms-radius in particular are the experiments of Simon et al., Berard et al. and Platchkov et al., which provide the most accurate cross sections at the lower q. In all cases we start from the original, unseparated cross sections. This allows us to make, as part of the fit of the data, the separation of the longitudinal and transverse structure functions using the full experimental information available today. For cases where the deuteron data have been measured relative to the proton, we recalculate the cross sections using more recent data for the proton [ 351. In our analysis we also take into account the dominant systematic errors (normalization, energy, ...) as quoted in the publications; in cases where these were incomplete we consulted the corresponding theses [ 36-391. The data basis used includes a total of 329 points.
18
I. Sick, D. Traumann /Physics Letters B 375 f 1996) 16-20
4. Previous analyses Before discussing results, it may be instructive to compare the present approach to the previous studies of Klarsfeld et al. and Wong. These authors have studied the low momentum transfer data [ 3 1,151 or individual sets of these; both studies relied on the published values of A(q). Klarsfeld et al. and Wong explored different parameterizations of the form factor, such as power series in q*, PadC approximants or sums of these with theoretical form factors. Klarsfeld et al. also determined some of the higher moments (r*“) (which for the deuteron are strongly correlated with the rms-radius), while Wong essentially disregarded them. The systematic errors of the data in general were only considered in part. The work of Refs. [ 1,2] has been performed in terms of the body form factor, which differs from the chauge form factor by the contribution of the proton and neutron finite size, and the Darwin-Foldy term. (According to [ 41 the MEC contribution is negligible at the level of interest here). In the present analysis we quote results in terms of the charge quantities, as these are the ones closest to the experimental data on electron scattering. The method employed here has several advantages: It uses the primary data (cross sections), it uses only the data to fix the higher moments of the density, and it employs only the safest part of theoretical knowledge on the deuteron wave function at large r-. Use of all data, which allows a very flexible parametrization, makes the results insensitive to the choice of the parameterization and to the cut-off in q.
5. Results of PWBA fits The fit of the world data leads to a charge rms-radius of the deuteron of 2.113 f 0.011 fm, where the error bar includes both statistical and systematic uncertainties of the data. This radius perfectly agrees with the one determined by Klarsfeld et al., 2.112 * 0.003 fm, despite the very different approach used. Our error bar is bigger, mainly as a consequence of the fact that we account for all the systematic errors of the data; these are taken into account by changing the data by their quoted error, refitting, and adding quadratically the changes resulting from the uncertainties of the differ-
ent experiments. Fits have also been performed without the tail-constraint mentioned above; we find the same radius within the larger error bar. The discrepancy with the radii derived from deuteron wave functions and atomic transitions thus at first sight seems to remain.
6. Coulomb distortion In the analysis described above, as well as in all previous studies of electron-deuteron scattering, the data have been interpreted in terms of the plane wave Born approximation (PWBA) .This approximation allows to express the cross section as a product of the Mott cross section and form factors, with the equations given above. Coulomb distortion was implicitly assumed to be small. However, Zcu N 1/ 137, the quantity which sets the scale of Coulomb corrections, is of order ? 1%. Corrections of this size cannot be ignored when one attempts to determine radii accurate to a fraction of a percent (the discrepancy of 0.0 19 fm we try to elucidate amounts to 0.9%). We have calculated the Coulomb distortion using second order Born approximation as described in [41,42]. For Za “v 0.01 this expansion in terms of powers of (Zcr) is expected to be very accurate, and we estimate it to give more reliable results than the standard phase-shift codes developed for treating the Coulomb distortion for heavy nuclei; the calculation in second order Born approximation is more transparent and more easily verified than the numerical solutions of the Dirac equation. The Coulomb corrections have been calculated using the known deuteron form factors. They change the form factors by up to 0.5%. This is a significant change given the fact that the finite size effect in the form factor, q*(r*)/6, is of order 0.3 at the momentum transfers where the form factors are the most sensitive to the rms-radius; 0.005 is not negligible as compared to 0.3. For absolute deuteron cross sections such as the ones of [ 28,3 1] the Coulomb corrections are straightforward. A complication could arise for those data [ 151 where the ratio deuteron/proton has been measured. In order to get absolute deuteron cross sections we multiply these ratios with the fit to the absolute cross sections [ 351 for the proton. Fits to the proton data always have also been done without considera-
I. Sick. D. Trautmann/Physics
tion of Coulomb distortion effects and therefore might be suspect. For the present purpose, the procedure we employ is adequate, however. We can simply consider these fits, although PWBA-inspired, as a convenient parameterization of the absolute proton experimental cross sections, and use them as such to recalculate the proton cross sections. If, on the other hand, one would want to determine the most accurate information on theproton charge density and radius, then one would have to account for the Coulomb distortion in electron-proton scattering; this is not the goal of the present Letter. We also note that a potential change of 1% of the proton radius due to Coulomb distortion affects little the theoretical radii we will compare to. In the calculated deuteron charge rms-radius this would lead to a change of 0.003 fm, a change which is much smaller than the uncertainty of the experimental deuteron radius.
7. Results After correcting the data for Coulomb distortion, the SOG tit to A(q) , B(q) yields a x2 of 1.3 per degree of freedom (the x2 improved by 20 relative to the fits ignoring Coulomb distortion). This good x2 is quite satisfactory for such a varied set of data taken over 30 years. The small systematic deviations we observe between individual sets of data and the fit indicate that the systematic uncertainties have not been overestimated. The deuteron rms-radius charge radius that results is 2.128 & 0.011 fm. When fitting, alternatively, Fcn( q), &I ((I), and Fc2 (q) we find a similar x2 per degree of freedom, and a similar radius, 2.126 & 0.012 fm. The value of the radius from the fit of A(q) , B(q) is probably a bit more reliable as this fit does not depend on the difficult separation of the CO and C2 form factors based on the very limited set of polarization observables today available. We find the same radius when omitting the Monterey-data [ 15,161, the only set of low-q data where the normalization to proton-data as discussed above comes in 2 . 2For optimum x2 it is necessary to renormalize the data by Berard by -0.4%. a change that agrees with the target contamination
Letters B 375 (1996) 16-20 Table 1 Deuteron rms-radii Klarsfeld Present, Present, Klarsfeld Pachucki
from different
19
sources
et al., from (e,e) PWIA with Coulomb distortion et al., from N-N potentials et al., 2p-2s transition
2.112 2.113 2.128 2.131 2.133
+ 0.003 fm i? 0.011 fm i 0.011 fm fm f 0.007 fm
The charge radius of the deuteron found in the present work, 2.128 f 0.011 fm, is in good agreement with the one derived from deuteron wave functions and the triplett scattering length [ I] which, expressed in terms of the charge radius, amounts to 2.131 * 0.003 fm, and the one obtained [f&5] from the 2p-2s transition in atomic deuterium, 2.133 5 0.007 fm.
8. Conclusions
In the present Letter we have analyzed the world data on electron-deuteron elastic scattering in order to determine the deuteron charge rms-radius. This study was undertaken in order to elucidate the long-standing discrepancy between the radii derived from electron scattering and other observables. We have shown that even for 2 = 1 it is important to take into account Coulomb distortion. When doing so the m-is-radii derived from deuteron wave functions and atomic transitions on the one side, and the one from electron scattering on the other, agree.
References S. Klarsfeld, J. Maaorell, J.A. Oteo, M. Nishimura and D.W.L. Sprung, Nucl. Phys. A 456 ( 1986) 373. I21 C.W. Wong, Int. J. Mod. Phys. 3 (1994) 821. 131 M. Kohno, J. of Physics G 9 (1983) L8S. [41 A. Buchmann, H. Henning and PU. Sauer, to be pub]., 1996. 151 E Schmidt-Kaler, D. Leibfried, M. Weitz and T.W. Haensch, Phys. Rev. Len. 70 ( 1993) 2261. 161 K. Pachucki, M. Weitz and T.W. Haensch, Phys. Rev. A 49 ( 1994) 2255. [71 R.G. Arnold, C.E. Carlson and E Gross, Phys. Rev. C 23 (1981) 363. 181 1. Sick, Nucl. Phys. A 218 ( 1974) 509. [91 H. Henning, private communication. IlO1 Yu.K. Akimov et al., Sov. J. Nucl. Phys. 29 (1979) 335. [Ill R.G. Arnold et al., Phys. Rev. Lett. 35 ( 1975) 776. [ 39
1 not taken into account in the published cross section ratios.
20
I. Sick, D. Traunnann/Physics
[ 12 1 R.G. Arnold et al., Phys. Rev. Lett. 58 (1987) 1723. 1131 S. Auffret, J.M. Cavedon, B. Frois, D. Goutte, M. Huet, Ph.
Lcconte, J. Martino, Y. Mizun, X.-H. Phan and S. Platchkov, Phys. Rev. Lett. 54 (1985) 649. 114 1 D. Benaksas, D. D&key and D. Frerejacque, Phys. Rev. 148 ( 1966) 1327. 1151 R.W. Berard, F.R. Buskirk, E.B. Dally, J.N. Dyer, X.K. Maruyama, R.L. Topping, and T.J. Traverso, Phys. Lett. B 47 (1973) 355. 1161 EA. Bumiller. ER. Buskirk, J.W. Stewart and E.B. Dally, Phys. Rev. Lett. 25 (1970) 1774. 117 1C.D. Buchanan and M.R. Yearian, Phys. Rev. Lett. 15 ( 1965) 303. [ 18 1 R. Cramer et al., Z. Phys. C 29 (1985) 513. ( 19 1 D. Drickey and L.N. Hand, Phys. Rev. Lett. 9 ( 1962) 521. I20 1J.E. Elias et al., Phys. Rev. 177 (1969) 2075. 12I 1 J.K. Friedman, H.W. Kendall and F!A.M. Gram, Phys. Rev. 120 (1960) 992. I22 1 S. Galster, H. Klein, J. Moritz, K.H. Schmidt, D. Wegener and J. Bleckwenn, Nucl. Phys. B 32 ( 1971) 221. I23 1 D. Ganichot, B. Grosset&e and D.B. Isabelle, Nucl. Phys. A 178 (1972) 545. [ 24 I R. Gilman et al., Phys. Rev. Lett. 65 (1990) 1733. I25 1 J. Goldemberg and C. Schaerf, Phys. Rev. lett. 12 ( 1964) 298. 1261 B. Grossetete, D. Drickey and P Lehmann, Phys. Rev. 141 (1966) 1425.
Letters B 375 (1996) 16-20
[27] E Martin et al., Phys. Rev. Len. 38 (1977) 1320. [28] S. Platchkov, A. Amroun, S. Auffret, J.M. Cavedon, P. Dreux, J. Duclos, B. Frois, D. Goutte, H. Hachemi, J. Mattino, X.H. Phan and 1. Sick, Nucl. Phys. A 510 (1990) 740. [ 291 R.E. Rand, RF. Frosch, C.E. Littig and M.R. Yearian, Phys. Rev. Len. I8 (1967) 469. 1301 M.E. Schulze et al., Phys. Rev. Len. 52 (1984) S97. [ 311 G.G. Simon, Ch. Schmitt and V.H. Walther, Nucl. Phys. A 364 (1981) 285. [32] P Stein, M. Binkley, R. McAllister, A. Suri and W. Woodward, Phys. Rev. Lett. 16 (1966) 592. (331 I. The et al., Phys. Rev. Lett. 67 (1991) 173. [34] B.B. Voitsekhovskii et al., Pis’ma Zh. Eksp. Teor. Fiz. 43 (1986) 567. [ 351 G. Hoehler, E. Pietarinen, I. Sabba-Stefanescu, E Borkowski, G.G. Simon, V.H. Walther and R.D. Wendling, Nucl. Phys. B 14 (1976) 505. [36] G.G. Simon, Dissertation, Johannes Gutenberg Universitlt Mainz, 1977. 1371 S. Auffret, These Universite Paris-Sud, Orsay 1985, 1985. [38] R.W. Berard, Thesis, Naval Postgraduate School Monterey, 1973. [ 39 1 R.L. Topping, Thesis, Naval Postgraduate School Monterey, 1972. 1401 R. Wilson, (Wiley, New York, 1963) p. 137. 1411 R.R. Lewis, Phys. Rev. 102 ( 1956) 537. 1421 SD. Drell and R.H. Pratt, Phys. Rev. 125 ( 1962) 1394.
cm
16 May 1996
PHYSICS
ELSEVIER
LETTERS B
Physics Letters B 375 ( 1996) 16-20
On the rms-radius of the deuteron I. Sick ‘, D. Trautmann Dept. fiir Physik und Astronomie, Universitiit Basel. CH-4056 Basel, Switzerland Received
16 January
1996; revised manuscript received 16 February Editor: R.H. Siemssen
1996
Abstract We study the world data on elastic electron-deuteron scattering in order to determine the deuteron charge rms-radius. After accounting for the Coulomb distortion we find a radius of 2.128f 0.011 fm, which is in perfect agreement with the
radii deduced from other sources. This removes a disturbing discrepancy first pointed out by Klarsfeld et al. Keywords: Deuteron
radius; Electron scattering;
Coulomb
distortion
1. Introduction The deuteron is a fundamental system that has received extensive attention in the past, by both theory and experiment. The wave function of the deuteron can be calculated accurately for any given nucleonnucleon (N-N) potential. Many observables, and the electromagnetic form factors in particular, can be predicted with high precision. Overall, the studies of the past have shown that the information on the deuteron emerging from NN scattering and the one from electron scattering are compatible. The two types of observables then can be used as complementary sources of knowledge on the properties of the 2N-system. The work of Klarsfeld et al. [ 11, however, has revealed a disturbing discrepancy: The deuteron root-mean-square (rms) radius calculated using many N-N potentials shows a tight linear correlation with the N-N triplett scattering length predicted by the potential employed. For the well known experimental scattering length this linear ’ E-mail: [email protected]. 0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PII SO370-2693(96)00214-6
relation misses the experimental deuteron rms-radius derived from electron scattering by 0.019 ho.003 fm. The very detailed treatise of Wong [2], which also gives reference to much of the previous work on the subject, has confirmed this disagreement For a system as well understood as the deuteron such a discrepancy is most disturbing. As a consequence, various authors have studied corrections that could remove this problem. In particular they have looked at the potential role played by an energydependence of the N-N interaction off-shell, and the role of non-nucleonic degrees of freedom. Mesonic [ 3,1,4J and quark [2,4] degrees have been investigated, but no explanation of the disagreement has been found; their effect is an order of magnitude too small. Recent very accurate data on the 2p-2s-transition energy in atomic deuterium [ 5,6] have drawn further attention to this question. Therms-radius derived from the atomic data is compatible with the one obtained from N-N scattering, but disagrees with the radius derived from electron scattering. In this Letter, we describe an alternative analysis of the electron scattering data on the deuteron which has
I. Sick, D. Trautmann/Physics
been performed in the hope to elucidate the origin of this puzzle.
2. Analysis In the present “model independent” analysis we write the elastic cross section in terms of the usual form factors F or structure functions A, B da(& 0) = ~Mott(Ev@)[A(q) + dQ
B(d$W/2>1
with the definition
+
Md ( xE.Ld)’ I’
B(q)
;+r&
= ( $%dj2
$T(
(9)
1+
rl)F;,(q)
with
I’ rl =
q2/(4MS)
and the corresponding equations for the tensor polarization observables [7]. In the above equations, M are the masses, ,u(Q) the deuteron magnetic (quadrupole) moment, and q is the four-momentum transfer. Of prime interest for this Letter is the low-q behaviour of FCQ N 1 - q2(r2)/6 + . . . . or the corresponding one for A(q) (corrected for the small FM] contribution using A (r*) = -&j/M;). We parameterize the deuteron form factors Fa (q) ,FM, (q) ,Fa (q) [or, alternatively, the structure functions A(q) and B(q)] using the sum-ofgaussians form factors SOG [ 81. This parameterization depends on one parameter, the width y of the gaussians which governs the large-q behaviour. This width is set to a value smaller than the size of the proton (we use 0.6 fm), as with increasing momentum transfer q the deuteron form factors are not expected to fall more slowly than the proton form factors, For the present study, only the behaviour of Fa (q) [or A(q) ] at small momentum transfer q is needed; the behaviour of the corresponding r-space densities a priori is of little interest. For a better understanding of a constraint we are going to use, it is useful, however, to also consider the SOG densities in r-space. (At low q the correspondence of density and form factor is well defined as the contribution of MEC is small).
Letters B 375 (1996116-20
17
The individual terms of the SOG basis correspond to gaussians of narrow width y placed at many different radii Ri. The deuteron wave function exhibits a long tail towards large I due to the weak binding of the deuteron, a tail that affects the shape of the form factors at very low q. As the shape of the tail at large r is well known - it depends only on the binding energy of the deuteron - this shape can be used as a constraint during the analysis of the data. This constraint amounts to fixing the ratio of the amplitudes of the gaussians for radii Ri 2 4 fm to the ratios given by a fit to the form factors or structure functions calculated from wave functions that have the correct binding energy. For the region r < 4 fm (where r refers to the radius relative to the center of mass of the deuteron) we use 10 parameters (gaussians) to describe each of the 3 (2) form factors. Various tests have been performed by fitting the form factors calculated by Henning et al. [9] for different N-N potentials. These tests show that the parameterization allows to fit the form factors up to the largest momentum transfers of the data, and they show that the fits accurately reproduce the rms-radii of the corresponding densities.
3. Data In the present work, we employ the world data on e-d scattering [ lo-341 ; relevant for a determination of the rms-radius in particular are the experiments of Simon et al., Berard et al. and Platchkov et al., which provide the most accurate cross sections at the lower q. In all cases we start from the original, unseparated cross sections. This allows us to make, as part of the fit of the data, the separation of the longitudinal and transverse structure functions using the full experimental information available today. For cases where the deuteron data have been measured relative to the proton, we recalculate the cross sections using more recent data for the proton [ 351. In our analysis we also take into account the dominant systematic errors (normalization, energy, ...) as quoted in the publications; in cases where these were incomplete we consulted the corresponding theses [ 36-391. The data basis used includes a total of 329 points.
18
I. Sick, D. Traumann /Physics Letters B 375 f 1996) 16-20
4. Previous analyses Before discussing results, it may be instructive to compare the present approach to the previous studies of Klarsfeld et al. and Wong. These authors have studied the low momentum transfer data [ 3 1,151 or individual sets of these; both studies relied on the published values of A(q). Klarsfeld et al. and Wong explored different parameterizations of the form factor, such as power series in q*, PadC approximants or sums of these with theoretical form factors. Klarsfeld et al. also determined some of the higher moments (r*“) (which for the deuteron are strongly correlated with the rms-radius), while Wong essentially disregarded them. The systematic errors of the data in general were only considered in part. The work of Refs. [ 1,2] has been performed in terms of the body form factor, which differs from the chauge form factor by the contribution of the proton and neutron finite size, and the Darwin-Foldy term. (According to [ 41 the MEC contribution is negligible at the level of interest here). In the present analysis we quote results in terms of the charge quantities, as these are the ones closest to the experimental data on electron scattering. The method employed here has several advantages: It uses the primary data (cross sections), it uses only the data to fix the higher moments of the density, and it employs only the safest part of theoretical knowledge on the deuteron wave function at large r-. Use of all data, which allows a very flexible parametrization, makes the results insensitive to the choice of the parameterization and to the cut-off in q.
5. Results of PWBA fits The fit of the world data leads to a charge rms-radius of the deuteron of 2.113 f 0.011 fm, where the error bar includes both statistical and systematic uncertainties of the data. This radius perfectly agrees with the one determined by Klarsfeld et al., 2.112 * 0.003 fm, despite the very different approach used. Our error bar is bigger, mainly as a consequence of the fact that we account for all the systematic errors of the data; these are taken into account by changing the data by their quoted error, refitting, and adding quadratically the changes resulting from the uncertainties of the differ-
ent experiments. Fits have also been performed without the tail-constraint mentioned above; we find the same radius within the larger error bar. The discrepancy with the radii derived from deuteron wave functions and atomic transitions thus at first sight seems to remain.
6. Coulomb distortion In the analysis described above, as well as in all previous studies of electron-deuteron scattering, the data have been interpreted in terms of the plane wave Born approximation (PWBA) .This approximation allows to express the cross section as a product of the Mott cross section and form factors, with the equations given above. Coulomb distortion was implicitly assumed to be small. However, Zcu N 1/ 137, the quantity which sets the scale of Coulomb corrections, is of order ? 1%. Corrections of this size cannot be ignored when one attempts to determine radii accurate to a fraction of a percent (the discrepancy of 0.0 19 fm we try to elucidate amounts to 0.9%). We have calculated the Coulomb distortion using second order Born approximation as described in [41,42]. For Za “v 0.01 this expansion in terms of powers of (Zcr) is expected to be very accurate, and we estimate it to give more reliable results than the standard phase-shift codes developed for treating the Coulomb distortion for heavy nuclei; the calculation in second order Born approximation is more transparent and more easily verified than the numerical solutions of the Dirac equation. The Coulomb corrections have been calculated using the known deuteron form factors. They change the form factors by up to 0.5%. This is a significant change given the fact that the finite size effect in the form factor, q*(r*)/6, is of order 0.3 at the momentum transfers where the form factors are the most sensitive to the rms-radius; 0.005 is not negligible as compared to 0.3. For absolute deuteron cross sections such as the ones of [ 28,3 1] the Coulomb corrections are straightforward. A complication could arise for those data [ 151 where the ratio deuteron/proton has been measured. In order to get absolute deuteron cross sections we multiply these ratios with the fit to the absolute cross sections [ 351 for the proton. Fits to the proton data always have also been done without considera-
I. Sick. D. Trautmann/Physics
tion of Coulomb distortion effects and therefore might be suspect. For the present purpose, the procedure we employ is adequate, however. We can simply consider these fits, although PWBA-inspired, as a convenient parameterization of the absolute proton experimental cross sections, and use them as such to recalculate the proton cross sections. If, on the other hand, one would want to determine the most accurate information on theproton charge density and radius, then one would have to account for the Coulomb distortion in electron-proton scattering; this is not the goal of the present Letter. We also note that a potential change of 1% of the proton radius due to Coulomb distortion affects little the theoretical radii we will compare to. In the calculated deuteron charge rms-radius this would lead to a change of 0.003 fm, a change which is much smaller than the uncertainty of the experimental deuteron radius.
7. Results After correcting the data for Coulomb distortion, the SOG tit to A(q) , B(q) yields a x2 of 1.3 per degree of freedom (the x2 improved by 20 relative to the fits ignoring Coulomb distortion). This good x2 is quite satisfactory for such a varied set of data taken over 30 years. The small systematic deviations we observe between individual sets of data and the fit indicate that the systematic uncertainties have not been overestimated. The deuteron rms-radius charge radius that results is 2.128 & 0.011 fm. When fitting, alternatively, Fcn( q), &I ((I), and Fc2 (q) we find a similar x2 per degree of freedom, and a similar radius, 2.126 & 0.012 fm. The value of the radius from the fit of A(q) , B(q) is probably a bit more reliable as this fit does not depend on the difficult separation of the CO and C2 form factors based on the very limited set of polarization observables today available. We find the same radius when omitting the Monterey-data [ 15,161, the only set of low-q data where the normalization to proton-data as discussed above comes in 2 . 2For optimum x2 it is necessary to renormalize the data by Berard by -0.4%. a change that agrees with the target contamination
Letters B 375 (1996) 16-20 Table 1 Deuteron rms-radii Klarsfeld Present, Present, Klarsfeld Pachucki
from different
19
sources
et al., from (e,e) PWIA with Coulomb distortion et al., from N-N potentials et al., 2p-2s transition
2.112 2.113 2.128 2.131 2.133
+ 0.003 fm i? 0.011 fm i 0.011 fm fm f 0.007 fm
The charge radius of the deuteron found in the present work, 2.128 f 0.011 fm, is in good agreement with the one derived from deuteron wave functions and the triplett scattering length [ I] which, expressed in terms of the charge radius, amounts to 2.131 * 0.003 fm, and the one obtained [f&5] from the 2p-2s transition in atomic deuterium, 2.133 5 0.007 fm.
8. Conclusions
In the present Letter we have analyzed the world data on electron-deuteron elastic scattering in order to determine the deuteron charge rms-radius. This study was undertaken in order to elucidate the long-standing discrepancy between the radii derived from electron scattering and other observables. We have shown that even for 2 = 1 it is important to take into account Coulomb distortion. When doing so the m-is-radii derived from deuteron wave functions and atomic transitions on the one side, and the one from electron scattering on the other, agree.
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